Dimension constraints in some problems involving intermediate curvature

Xu, Kai

doi:10.1090/tran/9332

Dimension constraints in some problems involving intermediate curvature
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by Kai Xu;

Trans. Amer. Math. Soc. 378 (2025), 2091-2112

DOI: https://doi.org/10.1090/tran/9332

Published electronically: November 19, 2024

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Abstract:

In [Comm. Pure Appl. Math. 77 (2024), pp. 441–456] Brendle-Hirsch-Johne proved that $T^m\times S^{n-m}$ does not admit metrics with positive $m$-intermediate curvature when $n\leq 7$. Chu-Kwong-Lee showed in [Math. Res. Lett., to appear] a corresponding rigidity statement when $n\leq 5$. In this paper, we show sharpness of the dimension constraints by giving concrete counterexamples in $n\geq 7$ and extending the rigidity result to $n=6$. Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension $\leq 5$ and bi-Ricci curvature $\geq 1$ have finite Urysohn 1-width. Counterexamples are constructed in dimension $\geq 6$.

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